Re: Basic concept I need explained for my calculus class

One way to thinking about this is to draw a line from the origin of a coordinate system to (2, 0) then down to (2, -1) then back to (0, 0). That gives you a right triangle with opposite side "-1" and "near side" 2 so that the tangent of the angle at the origin is -1/2. If you continue that line from the fourth quadrant to the second quadrant, you will get a line to (-2, 1) which gives a right triangle that has "near side" -2 and "opposite side" 1 so that we still have tangent equal to -1/2. That is why "tangent is periodic with period ". And of course, adding any multiple of to either of those will take you around in a complete circle and back to the same position.

Re: Basic concept I need explained for my calculus class

Well, if we just wanted to convert to polar coordinates, I would say:

(approximately). And for these conversions, you need to double-check: the point x=4, y=-2 is in quadrant 4, and r=4.472, theta=-0.464 is also in quadrant 4, and roughly the same place. I always draw a graph - that's hard to do here.

You might be aware that the polar representation of a given point is not unique. If is a point in the plane, then , , etc. represent the same point. And also , , etc.

In the problem, you are asked to find the two representations with angle between and . Since -0.464 is negative, that representation won't do. But if you add , you get r=4.472, theta=5.820 which is good, and the other one has r negative: r=-4.472, theta=2.678.